# Difference between revisions of "Locally compact space"

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− | A topological space at every point of which there is a neighbourhood with compact closure. A locally compact Hausdorff space | + | {{TEX|done}} |

+ | A topological space at every point of which there is a neighbourhood with compact closure. A locally compact Hausdorff space $X$ is a [[Completely-regular space|completely-regular space]]. The partially ordered set of all its Hausdorff compactifications (cf. [[Compactification|Compactification]]) is a complete lattice. Its minimal element is the [[Aleksandrov compactification|Aleksandrov compactification]] $\alpha X$. The class of locally compact Hausdorff spaces coincides with the class of open subsets of Hausdorff compacta. For a locally compact Hausdorff space $X$ its remainder $bX\setminus X$ in any Hausdorff compactification $bX$ is a Hausdorff compactum. Every connected paracompact locally compact space is the sum of countably many compact subsets. | ||

− | The most important example of a locally compact space is | + | The most important example of a locally compact space is $n$-dimensional Euclidean space. A topological Hausdorff [[Vector space|vector space]] $E$ (not reducing to the zero element) over a complete non-discretely normed division ring $k$ is locally compact if and only if $k$ is locally compact and $E$ is finite-dimensional over $k$. |

====Comments==== | ====Comments==== | ||

− | A product | + | A product $\prod_\alpha X_\alpha$ of topological spaces is locally compact if and only if each separate coordinate space $X_\alpha$ is locally compact and all but finitely many are compact. |

====References==== | ====References==== | ||

<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.L. Kelley, "General topology" , v. Nostrand (1955) pp. 146–147</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.L. Kelley, "General topology" , v. Nostrand (1955) pp. 146–147</TD></TR></table> |

## Revision as of 11:51, 6 July 2014

A topological space at every point of which there is a neighbourhood with compact closure. A locally compact Hausdorff space $X$ is a completely-regular space. The partially ordered set of all its Hausdorff compactifications (cf. Compactification) is a complete lattice. Its minimal element is the Aleksandrov compactification $\alpha X$. The class of locally compact Hausdorff spaces coincides with the class of open subsets of Hausdorff compacta. For a locally compact Hausdorff space $X$ its remainder $bX\setminus X$ in any Hausdorff compactification $bX$ is a Hausdorff compactum. Every connected paracompact locally compact space is the sum of countably many compact subsets.

The most important example of a locally compact space is $n$-dimensional Euclidean space. A topological Hausdorff vector space $E$ (not reducing to the zero element) over a complete non-discretely normed division ring $k$ is locally compact if and only if $k$ is locally compact and $E$ is finite-dimensional over $k$.

#### Comments

A product $\prod_\alpha X_\alpha$ of topological spaces is locally compact if and only if each separate coordinate space $X_\alpha$ is locally compact and all but finitely many are compact.

#### References

[a1] | J.L. Kelley, "General topology" , v. Nostrand (1955) pp. 146–147 |

**How to Cite This Entry:**

Locally compact space.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Locally_compact_space&oldid=32382